Triangulations and Simplicial Methods [Paperback]

As a new type of technique, simplicial methods have yielded extremely important contributions toward solutions of a system of nonlinear equations. Theoretical investigations and numerical tests have shown that the performance of simplicial methods depends critically on the triangulations underlying them. This monograph describes some recent developments in triangulations and simplicial methods. It includes the D1-triangulation and its applications to simplicial methods. As a result, efficiency of simplicial methods has been improved significantly. Thus more effective simplicial methods have been developed.As a new type of technique, simplicial methods have yielded extremely important contributions toward solutions of a system of nonlinear equations. Theoretical investigations and numerical tests have shown that the performance of simplicial methods depends critically on the triangulations underlying them. This monograph describes some recent developments in triangulations and simplicial methods. It includes the D1-triangulation and its applications to simplicial methods. As a result, efficiency of simplicial methods has been improved significantly. Thus more effective simplicial methods have been developed.1 Introduction.- 2 Preliminaries.- 2.1 Notations.- 2.2 Fixed Point Theorems.- 2.3 Applications.- 2.3.1 A Pure Exchange Economy.- 2.3.2 A Finite n-Person Normal Form Game.- 2.3.3 An Exchange Economy with Linear Production Technologies.- 2.3.4 Convex Programming.- 2.3.5 A Balanced Game without Side Payments.- 3 Existing Triangulations.- 3.1 Existing Triangulations of Sn.- 3.2 Existing Triangulations of Rn.- 3.3 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) ? Sn.- 3.4 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) ? Rn.- 4 The D1-Triangulation of Rn.- 4.1 The D1-Triangulation of Rn.- 4.2 Pivot Rules of the D1-Triangulation.- 4.3 The Number of Simplices of the D1-Triangulation in a Unit Cube.- 4.4 The Diameter of the lĮ